Galleries for root subsystems

TL;DR

This paper explores the projection and lifting of labelled galleries related to root subsystems, enabling the construction of topological embeddings of Bott-Samelson varieties that are skew equivariant and order-preserving.

Contribution

It introduces new methods for projecting and lifting labelled galleries, facilitating embeddings of Bott-Samelson varieties with specific equivariance and order properties.

Findings

Constructed topological embeddings of Bott-Samelson varieties

Established skew equivariance with respect to the compact torus

Demonstrated order-preserving properties on fixed point sets

Abstract

We consider projection and lifting of labelled galleries to and from roots subsystems. Our constructions allow us to construct some topological embeddings of Bott-Samelson varieties skew equivariant with respect to the compact torus and order-preserving on the sets of points fixed by it.